Not much moved by the plight of people in the aforementioned trailer, or their goals and personal traits, I was nonetheless intrigued by the freezing part of the excerpt. I searched for synopsis of the very episode and came across this paragraph (after Wikipedia):

The Dome begins to rotate, which shifts the colder air of the upper atmosphere downward, sending Chester’s Mill into a deep freeze.

“This looks like a job for the Nit-picking Man!” I said to myself, and proceeded to gather information.

First of all, I looked for some articles on the Dome in Google Scholar. To my surprise there was none, so I had to resort to this site as my main source on the topic.

The main premise for the story is as follows (my own summary):

There is a typical US town, blah, blah, blah, one day its citizens and few passers-by are cut off from the rest of the world by a giant impenetrable yet transparent force field or barrier that seems to be of unknown origin, blah, blah, blah. They are confused, many people die, their resources become scarce, and so on and so forth…

What about the size of the phenomenon? According to the series wiki page:

The Dome is estimated to be 10 miles in diameter and of unknown height.

So, in other words, the diameter equals about (10 * 1609m) 16090 metres or 16.09 km in SI units.

Great, I told myself, some data to crunch!

Some maths first: a dome is really a part of a sphere that has been sectioned with a plane. Take a look at this (crude) picture:

This is a sphere, r stands for its radius; all r’s are equal in length and they meet exactly in the center; if you combine a blue r with a red r you’ll get the diameter of the sphere.

Now, let’s cut our sphere into two identical halves. You just have to position your sectioning plane so that it’ll go through sphere’s center and apply a gentle mental push.

We now have a hemishpere – basically a dome that has its height equal to the radius of the base.

It’ll be our model of what the Dome from the TV series can look like. Let’s put Chester’s Mill inside and ponder about the dimensions.

Height is half the diameter, so it’s a little bit more than 8 kilometres.

Now we need some meteo stuff. Earth’s atmosphere is not a homogenous gas cloud around our planet, but rather a layered cake of different densities and temperatures.

The lowest and most dense part is the troposphere – a layer of air about 17 km thick (on average – it’s much thinner above the Polar regions and a bit deeper around the Equator). We live just at its bottom and breathe it every day.

Because the higher we get in the troposphere, the less dense the air becomes, it gets colder and colder in the process. We can approximate that the temperature drops roughly 6.5 °C for each kilometre that we go up. This varies if the air contains little or no water vapor, but since landscape in the series does not resemble Sahara, let’s stick to humid air coefficient.

Now, some more maths and another hastily painted picture:

If we went up to the highest point of the sphere this temperature drop would be 52.29°C (8.045 km * 6.5 °C).

Series takes place in Maine, US. I assume it’s the middle of summer. According to this site, temperatures in this state are in range from 15 to 27 °C in July. Let’s average that to 21 °C.

It would mean that at 8 km up from the center of the Dome the air temperature drops to -31.29°C (21-52.29). That’s pretty much fracking freezing if you asked me.